Theorem mircgrextend 28627

Description: Link congruence over a pair of mirror points. cf tgcgrextend 28430. (Contributed by Thierry Arnoux, 4-Oct-2020.)

Hypotheses

Ref	Expression
mirval.p	⊢ 𝑃 = (Base‘𝐺)
mirval.d	⊢ − = (dist‘𝐺)
mirval.i	⊢ 𝐼 = (Itv‘𝐺)
mirval.l	⊢ 𝐿 = (LineG‘𝐺)
mirval.s	⊢ 𝑆 = (pInvG‘𝐺)
mirval.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
mirtrcgr.e	⊢ ∼ = (cgrG‘𝐺)
mirtrcgr.m	⊢ 𝑀 = (𝑆‘𝐵)
mirtrcgr.n	⊢ 𝑁 = (𝑆‘𝑌)
mirtrcgr.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
mirtrcgr.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
mirtrcgr.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
mirtrcgr.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
mircgrextend.1	⊢ (𝜑 → (𝐴 − 𝐵) = (𝑋 − 𝑌))

Assertion

Ref	Expression
mircgrextend	⊢ (𝜑 → (𝐴 − (𝑀‘𝐴)) = (𝑋 − (𝑁‘𝑋)))

Proof of Theorem mircgrextend

Step	Hyp	Ref	Expression
1		mirval.p	. 2 ⊢ 𝑃 = (Base‘𝐺)
2		mirval.d	. 2 ⊢ − = (dist‘𝐺)
3		mirval.i	. 2 ⊢ 𝐼 = (Itv‘𝐺)
4		mirval.g	. 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5		mirtrcgr.a	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
6		mirtrcgr.b	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
7		mirval.l	. . 3 ⊢ 𝐿 = (LineG‘𝐺)
8		mirval.s	. . 3 ⊢ 𝑆 = (pInvG‘𝐺)
9		mirtrcgr.m	. . 3 ⊢ 𝑀 = (𝑆‘𝐵)
10	1, 2, 3, 7, 8, 4, 6, 9, 5	mircl 28606	. 2 ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑃)
11		mirtrcgr.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
12		mirtrcgr.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
13		mirtrcgr.n	. . 3 ⊢ 𝑁 = (𝑆‘𝑌)
14	1, 2, 3, 7, 8, 4, 12, 13, 11	mircl 28606	. 2 ⊢ (𝜑 → (𝑁‘𝑋) ∈ 𝑃)
15	1, 2, 3, 7, 8, 4, 6, 9, 5	mirbtwn 28603	. . 3 ⊢ (𝜑 → 𝐵 ∈ ((𝑀‘𝐴)𝐼𝐴))
16	1, 2, 3, 4, 10, 6, 5, 15	tgbtwncom 28433	. 2 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼(𝑀‘𝐴)))
17	1, 2, 3, 7, 8, 4, 12, 13, 11	mirbtwn 28603	. . 3 ⊢ (𝜑 → 𝑌 ∈ ((𝑁‘𝑋)𝐼𝑋))
18	1, 2, 3, 4, 14, 12, 11, 17	tgbtwncom 28433	. 2 ⊢ (𝜑 → 𝑌 ∈ (𝑋𝐼(𝑁‘𝑋)))
19		mircgrextend.1	. 2 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝑋 − 𝑌))
20	1, 2, 3, 4, 5, 6, 11, 12, 19	tgcgrcomlr 28425	. . 3 ⊢ (𝜑 → (𝐵 − 𝐴) = (𝑌 − 𝑋))
21	1, 2, 3, 7, 8, 4, 6, 9, 5	mircgr 28602	. . 3 ⊢ (𝜑 → (𝐵 − (𝑀‘𝐴)) = (𝐵 − 𝐴))
22	1, 2, 3, 7, 8, 4, 12, 13, 11	mircgr 28602	. . 3 ⊢ (𝜑 → (𝑌 − (𝑁‘𝑋)) = (𝑌 − 𝑋))
23	20, 21, 22	3eqtr4d 2779	. 2 ⊢ (𝜑 → (𝐵 − (𝑀‘𝐴)) = (𝑌 − (𝑁‘𝑋)))
24	1, 2, 3, 4, 5, 6, 10, 11, 12, 14, 16, 18, 19, 23	tgcgrextend 28430	1 ⊢ (𝜑 → (𝐴 − (𝑀‘𝐴)) = (𝑋 − (𝑁‘𝑋)))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107  ‘cfv 6541  (class class class)co 7413  Basecbs 17230  distcds 17283  TarskiGcstrkg 28372  Itvcitv 28378  LineGclng 28379  cgrGccgrg 28455  pInvGcmir 28597

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7370  df-ov 7416  df-trkgc 28393  df-trkgb 28394  df-trkgcb 28395  df-trkg 28398  df-mir 28598

This theorem is referenced by:  mirtrcgr  28628